Coordinate Proofs

Grade 9 · geometry · 26 practice problems · read aloud

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What is a Coordinate Proof? 📐

A coordinate proof uses a coordinate plane and algebra to prove geometric concepts. Instead of just words and diagrams, you place shapes on a grid and use formulas (like distance, slope, and midpoint) to show that properties are true. It's useful because it gives you a precise, mathematical way to prove things about shapes.

How to Solve Problems: A Step-by-Step Guide

  1. Place the Figure: Position the shape on the coordinate plane in a way that makes calculations easy. Often, you'll place a vertex at the origin (0,0) and a side along the x-axis.
  2. Label the Coordinates: Write down the coordinates for all the key points (vertices, midpoints).
  3. State the Goal: What are you trying to prove? (e.g., "Prove two sides are equal.")
  4. Apply Formulas: Use the distance, slope, and midpoint formulas to calculate what you need.
  5. Draw a Conclusion: Compare your results to prove the statement.

Visual Examples

Example 1: Prove a Triangle is Isosceles

Given: Triangle ABC with A(0,0), B(6,0), C(3,4). Prove: It is isosceles.

  1. We need to show two sides are equal. Let's find AC and BC.
  2. Use the Distance Formula: √[(x₂-x₁)² + (y₂-y₁)²]
  3. AC = √[(3-0)² + (4-0)²] = √(9+16) = √25 = 5
  4. BC = √[(3-6)² + (4-0)²] = √(9+16) = √25 = 5
  5. Since AC = BC = 5, triangle ABC is isosceles. ✅

Example 2: Prove a Quadrilateral is a Parallelogram

Given: Quadrilateral with vertices W(0,0), X(4,0), Y(5,3), Z(1,3). Prove: It's a parallelogram.

  1. A parallelogram has opposite sides that are parallel (equal slopes).
  2. Find slopes using: m = (y₂-y₁)/(x₂-x₁)
  3. Slope WX = (0-0)/(4-0) = 0. Slope ZY = (3-3)/(5-1) = 0. So WX ∥ ZY.
  4. Slope WZ = (3-0)/(1-0) = 3. Slope XY = (3-0)/(5-4) = 3. So WZ ∥ XY.
  5. Both pairs of opposite sides are parallel, so it's a parallelogram. ✅

Common Mistakes to Avoid

Poor Placement: Placing the shape randomly, making calculations messy. Fix: Always put a vertex at (0,0) and a side on an axis.

Formula Errors: Mixing up x and y in the distance or slope formulas. Fix: Write the formula clearly and substitute carefully.

Assuming, Not Proving: Saying a shape "looks like" a parallelogram. Fix: You MUST use calculations to prove it.

Tips & Tricks

  • Right Angles: Perpendicular lines have slopes that are negative reciprocals (e.g., 2 and -1/2).
  • Midpoints: The midpoint formula averages the x's and averages the y's.
  • Strategy: Use distance formula for congruent sides. Use slope formula for parallel/perpendicular sides.

How to Practice

  • Start by proving simple things, like showing a triangle is right-angled.
  • Draw your own shapes on graph paper and prove their properties.
  • Practice using all three formulas: distance, slope, and midpoint.

Practice problems

6 of the 26, worked through step by step — try them before opening the answer.

1 Prove that triangle Hana with vertices A(2,2), B(6,2), C(4,6) is isosceles.

Hint: Use the distance formula to calculate the lengths of all three sides and compare them to determine if any two sides are equal.

Show the answer

Answer: AB = 4, AC = sqrt(20), BC = sqrt(20), AC = BC

  1. Calculate AB using distance formula: sqrt((6-2)^2 + (2-2)^2) = sqrt(16 + 0) = sqrt(16) = 4
  2. Calculate AC using distance formula: sqrt((4-2)^2 + (6-2)^2) = sqrt(4 + 16) = sqrt(20)
  3. Calculate BC using distance formula: sqrt((4-6)^2 + (6-2)^2) = sqrt(4 + 16) = sqrt(20)
  4. Compare side lengths: AB = 4, AC = sqrt(20), BC = sqrt(20)
  5. Since AC = BC, triangle Hana is isosceles with two equal sides.

2 Prove that triangle ABC with vertices A(1,1), B(5,1), C(3,7) is isosceles using coordinate geometry.

Hint: Calculate the distances between all three pairs of vertices using the distance formula. If two sides have equal length, the triangle is isosceles.

Show the answer

Answer: Triangle ABC is isosceles because AB = BC = 4 units, while AC = 2√10 units

An isosceles triangle has at least two sides of equal length. Using the distance formula, we can calculate the lengths of all three sides and compare them. The distance formula is d = √[(x₂-x₁)² + (y₂-y₁)²].

3 Prove that triangle Hana with vertices H(0,0), A(6,0), N(2,4) is isosceles using coordinate geometry.

Hint: Use the distance formula to calculate the lengths of all three sides and compare them. Remember that an isosceles triangle has at least two equal sides.

Show the answer

Answer: The triangle is isosceles because sides HN and AN both have length sqrt(20)

To prove a triangle is isosceles using coordinate geometry, calculate the distances between all pairs of vertices using the distance formula. If two sides have equal lengths, the triangle is isosceles. The distance formula is d = sqrt((x₂-x₁)² + (y₂-y₁)²).

4 Prove that triangle Emma with vertices E(3,1), M(9,1), M(6,7) is isosceles using coordinate geometry.

Hint: Use the distance formula to calculate the lengths of all three sides and compare them.

Show the answer

Answer: The triangle is isosceles because EM = MM = sqrt(45)

An isosceles triangle has at least two sides of equal length. The distance formula helps determine side lengths from coordinates.

5 Prove that triangle Emma with vertices E(1,1), M(5,1), M(3,5) is isosceles using coordinate geometry.

Hint: Use the distance formula to calculate the lengths of all three sides and compare them to determine if any two sides are equal.

Show the answer

Answer: The triangle is isosceles because EM = MM = 4 units

An isosceles triangle has at least two sides of equal length. The distance formula helps calculate side lengths using coordinates. If two sides have the same length, the triangle is isosceles.

6 Prove that triangle Aroha with vertices A(1,1), B(5,3), C(3,7) is isosceles using coordinate geometry.

Hint: Use the distance formula to calculate the lengths of all three sides and compare them. Remember that an isosceles triangle has at least two equal sides.

Show the answer

Answer: Triangle Aroha is isosceles because AB = BC = sqrt(20)

To prove a triangle is isosceles using coordinate geometry, calculate the distances between all three pairs of vertices using the distance formula. If any two sides have equal lengths, the triangle is isosceles. The distance formula is d = sqrt((x₂-x₁)² + (y₂-y₁)²).

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